Exam Study Notes

Univariate Analysis

  • Useful for describing the characteristics of a single variable.
  • Provides summaries about:
    • Central tendency (where data clusters).
    • Dispersion (how spread out data is).
  • Common measures:
    • Mean
    • Median
    • Mode
    • Standard Deviation
    • Variance
    • Skewness
    • Kurtosis
    • Frequencies
    • Percentages
  • How to perform in SPSS:
    • For categorical data: Analyze > Descriptive Statistics > Frequencies…
    • Select variables.
    • Check 'Display frequency tables'.
    • For charts: Charts… (Bar charts, Pie charts).
    • For continuous data: Analyze > Descriptive Statistics > Descriptives…
    • Select variables.
    • Click Options… to select mean, std. deviation, min, max, skewness, kurtosis.
    • Explore (for more detailed exploration and normality checks): Analyze > Descriptive Statistics > Explore…
    • Select variables for 'Dependent List'.
    • Plots… to get Boxplots, Histograms, Stem-and-leaf, and Normality plots with tests (Kolmogorov-Smirnov, Shapiro-Wilk).
  • How to interpret (SPSS Output):
    • Frequency Tables:
    • Look at the 'Frequency' and 'Percent' columns to understand distribution for categorical variables.
    • 'Valid Percent' is important if you have missing data.
    • Descriptives Table:
    • Mean: Average value.
    • Std. Deviation: How much on average individual data points deviate from the mean. A larger std. dev. means data is more spread out.
    • Skewness: Indicates the asymmetry of the distribution.
    • Positive skew: Tail to the right (data clustered on the left).
    • Negative skew: Tail to the left (data clustered on the right).
    • Rule of thumb: If Skewness/Std.ErrorofSkewness>±1.96Skewness / Std. Error of Skewness > ±1.96 (for large samples), it's significantly skewed.
    • Kurtosis: Indicates the 'peakedness' or 'flatness' of the distribution.
    • Positive kurtosis (leptokurtic): More peaked, heavier tails (more outliers).
    • Negative kurtosis (platykurtic): Flatter, lighter tails.
    • Rule of thumb: If Kurtosis/Std.ErrorofKurtosis>±1.96Kurtosis / Std. Error of Kurtosis > ±1.96 (for large samples), it's significantly kurtotic.
    • Histograms/Boxplots: Visually assess distribution, skewness, and presence of outliers.
    • Normality Tests (from Explore):
    • Kolmogorov-Smirnov (K-S) and Shapiro-Wilk: If the 'Sig.' value (p-value) is

Bivariate Analysis

  • Useful for: Examining the relationship between two variables.
  • How to perform in SPSS: Depends on variable types (see specific tests below: Chi-Square for categorical-categorical, T-test/ANOVA for categorical-continuous, Correlation/Regression for continuous-continuous).
  • How to interpret: The interpretation is specific to the test used.

Standard Deviation

  • Useful for: Quantifying the amount of variation or dispersion of a set of data values. It's the square root of the variance.
  • A low standard deviation indicates that the data points tend to be close to the mean of the set.
  • A high standard deviation indicates that the data points are spread out over a wider range of values.
  • How to perform in SPSS: Automatically calculated in Analyze > Descriptive Statistics > Descriptives… or Analyze > Descriptive Statistics > Frequencies… (under Statistics button).
  • How to interpret (SPSS Output):
    • Descriptive Statistics or Statistics Table: Look at the 'Std. Deviation' row/column.
    • Interpretation:
    • Compare it to the mean
    • A standard deviation that is a large proportion of the mean suggests high variability.
    • If comparing two groups, the group with a larger standard deviation has more spread-out data.

Comparing Two Means (Independent Samples T-test)

  • Useful for: Determining if there is a statistically significant difference between the means of two independent groups on a continuous variable. (e.g., Is there a difference in spending between male and female customers?)
  • Assumptions:
  1. Independent observations: Data from one group doesn't influence the other.
  2. Continuous dependent variable: The variable you're measuring (e.g., spending) is continuous.
  3. Normally distributed dependent variable: In each group (less critical with larger sample sizes due to Central Limit Theorem).
  4. Homogeneity of variances: The variance of the dependent variable is roughly equal in both groups (checked by Levene's Test).
  • How to perform in SPSS:
    • Analyze > Compare Means > Independent-Samples T Test…
    • Move your continuous variable (e.g., 'Spending') to the 'Test Variable(s):' box.
    • Move your categorical grouping variable (e.g., 'Gender') to the 'Grouping Variable:' box.
    • Click Define Groups… and enter the numerical codes for your two groups (e.g., 1 for Male, 2 for Female). Click Continue.
    • Click OK.
  • How to interpret (SPSS Output):
  1. Group Statistics Table:
    • Look at: 'Mean', 'N', 'Std. Deviation' for each group. This gives you a descriptive overview.
  2. Independent Samples Test Table: This is where the crucial interpretation happens.
    • A. Levene's Test for Equality of Variances: (First two columns of the table).
    • Sig. (p-value): If Sig.

Levene's Test

  • Useful for: Specifically checking the assumption of homogeneity of variances (equal variances) required for Independent Samples T-tests and ANOVA.
  • How to perform in SPSS: It's automatically included in the output when you run an Independent Samples T-test or a One-Way ANOVA.
  • How to interpret (SPSS Output):
    • Where to find: In the Independent Samples Test table (for t-test) or Test of Homogeneity of Variances table (for ANOVA).
    • Sig. (p-value):
    • If Sig.

ANOVA (One-Way Analysis of Variance)

  • Useful for: Determining if there is a statistically significant difference between the means of three or more independent groups on a continuous variable. (e.g., Is there a difference in customer satisfaction across three different product designs?)
  • Assumptions:
  1. Independent observations.
  2. Continuous dependent variable.
  3. Normally distributed dependent variable: In each group (less critical with larger sample sizes).
  4. Homogeneity of variances: Variances of the dependent variable are roughly equal in all groups (checked by Levene's Test).
  • How to perform in SPSS:
    • Analyze > Compare Means > One-Way ANOVA…
    • Move your continuous variable (e.g., 'Satisfaction') to the 'Dependent List:' box.
    • Move your categorical grouping variable with 3+ groups (e.g., 'Product Design') to the 'Factor:' box.
    • Post Hoc: Click Post Hoc… and select appropriate tests (e.g., Tukey if variances are equal, Games-Howell if not).
    • Options: Click Options… and check 'Descriptive', 'Homogeneity of variance test' (for Levene's), and possibly 'Means plot'.
    • Click OK.
  • How to interpret (SPSS Output):
  1. Descriptives Table: Provides means, standard deviations, etc., for each group. Good for initial overview.
  2. Test of Homogeneity of Variances Table (Levene's Test):
    • Sig. (p-value): Interpret as described in Levene's Test section. If Sig.

Post Hoc Test

  • Useful for: Used only when an ANOVA's main F-test is statistically significant. They pinpoint exactly which specific group means differ from each other. Without them, you only know a difference exists, not where.
  • How to perform in SPSS: Selected under Post Hoc… options in the One-Way ANOVA dialog.
    • Choosing a test:
    • Tukey HSD: Most common, good for equal sample sizes, assumes equal variances.
    • Bonferroni: More conservative (less likely to find false positives), good for unequal sample sizes, assumes equal variances.
    • Games-Howell: Recommended when Levene's Test is significant (variances are not equal).
  • How to interpret (SPSS Output):
    • Multiple Comparisons Table: (For Tukey, Bonferroni, etc.)
    • This table shows pairwise comparisons between all possible pairs of groups.
    • Mean Difference (I-J): The difference between the means of the two groups being compared.
    • Sig. (p-value): This is the crucial part.
    • If Sig.

Correlations (Pearson and Spearman)

  • Measuring the strength and direction of a linear relationship between two continuous variables.
  • Pearson's r: For normally distributed continuous data.
  • Spearman's rho: For ordinal data or continuous data that is not normally distributed (non-parametric).
  • Interpretation of Correlation Coefficient (r or rho):
    • Direction:
    • Positive (+): As one variable increases, the other also increases.
    • Negative (-): As one variable increases, the other decreases.
    • Strength (absolute value):
    • 0 to ± 0.1: Very weak/negligible
    • ± 0.1 to ± 0.3: Weak
    • ± 0.3 to ± 0.5: Moderate
    • ± 0.5 to ± 0.7: Strong
    • ± 0.7 to ± 1.0: Very Strong
  • Assumptions (Pearson):
  1. Continuous variables.
  2. Linear relationship: The relationship should look roughly linear when plotted.
  3. No significant outliers.
  4. Normally distributed variables: (less critical for the correlation coefficient itself, but relevant for significance testing).
  • How to perform in SPSS:
    • Analyze > Correlate > Bivariate…
    • Move your two (or more) continuous variables to the 'Variables:' box.
    • Check 'Pearson' (default) for parametric or 'Spearman' for non-parametric.
    • Ensure 'Flag significant correlations' is checked.
    • Click OK.
  • How to interpret (SPSS Output):
    • Correlations Table:
    • Pearson Correlation or Spearman Correlation row/column: This is your correlation coefficient (r or rho).
    • Sig. (2-tailed) (p-value):
    • If Sig.

Simple Linear Regression

  • Predicting the value of a continuous dependent variable (Y) based on the value of a single continuous independent variable (X). It allows you to model the linear relationship and make predictions.
  • Equation: Y=b<em>0+b</em>1X+eY = b<em>0 + b</em>1X + e (where b<em>0b<em>0 is the intercept, b</em>1b</em>1 is the slope, e is error).
  • Assumptions:
  1. Linearity: The relationship between X and Y is linear.
  2. Independence of residuals: Errors are independent.
  3. Homoscedasticity: The variance of the residuals is constant across all levels of X.
  4. Normality of residuals: Residuals are normally distributed.
  5. No significant outliers.
  • How to perform in SPSS:
    • Analyze > Regression > Linear…
    • Move your continuous dependent variable (the one you want to predict, e.g., 'Sales') to the 'Dependent:' box.
    • Move your continuous independent variable (the predictor, e.g., 'Advertising_Spending') to the 'Independent(s):' box.
    • Statistics…: Check 'Estimates', 'Model fit', 'Descriptives', 'Part and partial correlations', 'Collinearity diagnostics', 'Durbin-Watson'.
    • Plots…: To check assumptions. Put *ZRESID on Y-axis, *ZPRED on X-axis (for homoscedasticity). Also, Histogram and Normal probability plot for residuals.
    • Click OK.
  • How to interpret (SPSS Output):
  1. Model Summary Table:
    • R: Pearson correlation coefficient between observed and predicted values (same as correlation if only one predictor).
    • R-square (R2R^2): Proportion of variance in the dependent variable explained by the independent variable(s).
    • Example: R2R^2 = 0.60 means 60% of the variation in 'Sales' can be explained by 'Advertising Spending'. Higher is generally better, but can be inflated with more predictors.
    • Adjusted R-square: A more conservative estimate of the population R2R^2, especially useful in multiple regression.
    • Std. Error of the Estimate: The typical distance between observed and predicted values.
    • Durbin-Watson: Checks for autocorrelation of residuals (values near 2 are good, indicating no autocorrelation).
  2. ANOVA Table:
    • Sig. (p-value): Tests the overall significance of the regression model.
    • If Sig.

Multiple Regression

  • Predicting the value of a continuous dependent variable (Y) based on the values of two or more continuous or categorical independent variables (X<em>1X<em>1, X</em>2X</em>2, etc.). It allows you to see the unique contribution of each predictor while controlling for others.
  • Equation: Y=b<em>0+b</em>1X<em>1+b</em>2X<em>2+⋯+b</em>kXk+eY = b<em>0 + b</em>1X<em>1 + b</em>2X<em>2 + \cdots + b</em>kX_k + e
  • Assumptions: Same as simple regression, plus:
    • No multicollinearity: Independent variables should not be too highly correlated with each other (checked by VIF/Tolerance).
  • How to perform in SPSS:
    • Analyze > Regression > Linear…
    • Move your continuous dependent variable to the 'Dependent:' box.
    • Move all your independent variables (continuous or dummy-coded categorical) to the 'Independent(s):' box.
    • Statistics…: Essential to check 'Collinearity diagnostics' (for VIF/Tolerance). Also 'Estimates', 'Model fit', 'R squared change' (if doing hierarchical), 'Descriptives', 'Durbin-Watson'.
    • Plots…: Same as simple regression (*ZRESID on Y, *ZPRED on X).
    • Click OK.
  • How to interpret (SPSS Output):
  1. Model Summary Table:
    • R-square (R2R^2): Total proportion of variance in the dependent variable explained by all independent variables in the model.
    • Adjusted R-square: Key for multiple regression, as it adjusts for the number of predictors and gives a better estimate of the population R2R^2. Always report this in multiple regression.
    • Durbin-Watson: Check for autocorrelation.
  2. ANOVA Table:
    • Sig. (p-value): Overall significance of the entire regression model.
    • If Sig.

Chi-Square Test for Independence (χ2\chi^2)

  • Determining if there is a statistically significant association (relationship) between two categorical variables. (e.g., Is there a relationship between gender and preferred soda brand?). It tests if observed frequencies differ significantly from expected frequencies (what you'd expect if there was no association).
  • Assumptions:
  1. Categorical variables: Both variables are nominal or ordinal.
  2. Independent observations.
  3. Expected cell counts: No more than 20% of cells should have an expected count less than 5, and no cell should have an expected count less than 1. If violated, consider combining categories or using Fisher's Exact Test.
  • How to perform in SPSS:
    • Analyze > Descriptive Statistics > Crosstabs…
    • Move one categorical variable to 'Row(s):' and the other to 'Column(s):'.
    • Click Statistics…: Check 'Chi-square'.
    • Click Cells…: Check 'Observed' and 'Expected' counts. Also 'Row', 'Column', 'Total' percentages if desired.
    • Click OK.
  • How to interpret (SPSS Output):
  1. Crosstabulation Table:
    • Shows the observed counts and percentages for each cell (combination of categories). This is your raw data display.
    • Check the 'Expected Count' row/column to ensure assumption of expected cell counts is met.
  2. Chi-Square Tests Table: This is the main table for results.
    • Pearson Chi-Square row:
    • Value: The calculated Chi-Square statistic.
    • df: Degrees of freedom.
    • Asymp. Sig. (2-sided) (p-value):
    • If Sig.

Factor Analysis (Exploratory Factor Analysis - EFA)

  • A data reduction technique. It identifies underlying, unobserved "factors" or "dimensions" that explain the correlations among a larger set of observed variables. Used to simplify complex data and identify latent constructs (e.g., identifying different dimensions of "customer satisfaction" from many survey questions).
  • Key Concepts:
    • Factors: The underlying latent constructs.
    • Loadings: Correlation between an observed variable and a factor (how strongly a variable relates to a factor).
    • Eigenvalues: Amount of variance explained by each factor.
    • Communality: Proportion of variance in an observed variable explained by the extracted factors.
  • Assumptions:
  1. Continuous variables: Or at least interval/ratio.
  2. Adequate sample size: Generally, at least 5-10 observations per variable, or N > 100-200.
  3. Sufficient correlations: Variables should be sufficiently correlated with each other for factors to emerge (checked by KMO and Bartlett's Test).
  • How to perform in SPSS:
    • Analyze > Dimension Reduction > Factor…
    • Move all relevant continuous variables (e.g., survey items on satisfaction) to the 'Variables:' box.
    • Descriptives…: Check 'KMO and Bartlett's test of sphericity', 'Reproduced', 'Anti-image'.
    • Extraction…:
    • Method: 'Principal Components' (default, often used) or 'Principal Axis Factoring' (if interested in latent constructs).
    • Principal Components Analysis (PCA): This method aims to explain the total variance in the data. It seeks to find components that capture the maximum variance in the observed variables. PCA is often used as a first step in factor analysis to reduce the number of variables.
    • Principal Axis Factoring (PAF): This method aims to explain the covariance among the variables. It seeks to find factors that underlie the observed variables, focusing on the common variance. PAF is used when the primary goal is to uncover latent constructs.
    • Extract: 'Eigenvalues greater than 1' (Kaiser criterion) or specify number of factors.
    • Display: 'Unrotated factor solution' and 'Scree plot'.
    • Rotation…:
    • Method: 'Varimax' (orthogonal, assumes factors are uncorrelated) or 'Direct Oblimin' (oblique, allows factors to be correlated, often more realistic). Choose Varimax unless you have a strong theoretical reason for correlated factors.
    • Varimax: An orthogonal rotation method that maximizes the variance of the loadings within each factor. This tends to produce factors that are easier to interpret because each variable tends to load highly on one factor and less so on the others. It assumes that the factors are uncorrelated.
    • Direct Oblimin: An oblique rotation method that allows factors to be correlated. It is more realistic than Varimax when there is a theoretical reason to believe that the factors are related. Direct Oblimin can provide a simpler factor structure and is particularly useful when factors are expected to influence each other.
    • Display: 'Rotated solution'.
    • Options…: Check 'Sorted by size' and 'Suppress small coefficients' (e.g., absolute value below 0.3 or 0.4).
    • Click OK.
  • How to interpret (SPSS Output):
  1. KMO and Bartlett&